Let `f(x)=2-|x-3|,1<= x <= 5` and for rest of the values f(x) can be obtained by using the relation `f(5x)=alpha f(x) AA x in R` The maximum value of `f(x)` in `[5^4,5^5]` for `alpha=5` is

Options

A. 1118
B. 2007
C. 1250
D. 132

Answer

To find the maximum value of the function \( f(x) = 2 - |x - 3| \) for \( 1 \leq x \leq 5 \) and for \( x \) in the range \( [5^4, 5^5] \) using the relation \( f(5x) = \alpha f(x) \) where \( \alpha = 5 \), we will follow these steps:

### Step 1: Analyze the function \( f(x) \) for \( 1 \leq x \leq 5 \)
The function \( f(x) = 2 - |x - 3| \) is defined piecewise based on the value of \( x \):
- For \( x < 3 \): \( f(x) = 2 - (3 - x) = x - 1 \)
- For \( x = 3 \): \( f(3) = 2 - |3 - 3| = 2 \)
- For \( x > 3 \): \( f(x) = 2 - (x - 3) = 5 - x \)

### Step 2: Find the maximum value of \( f(x) \) in the interval \( [1, 5] \)
- At \( x = 1 \): \( f(1) = 1 - 1 = 0 \)
- At \( x = 2 \): \( f(2) = 2 - 1 = 1 \)
- At \( x = 3 \): \( f(3) = 2 \)
- At \( x = 4 \): \( f(4) = 5 - 4 = 1 \)
- At \( x = 5 \): \( f(5) = 5 - 5 = 0 \)

The maximum value of \( f(x) \) in the interval \( [1, 5] \) is \( 2 \) at \( x = 3 \).

### Step 3: Use the relation \( f(5x) = \alpha f(x) \) to find \( f(x) \) for \( x \) in \( [5^4, 5^5] \)
Given \( \alpha = 5 \), we can express \( f(5^4) \) and \( f(5^5) \):
- Let \( x = 3 \) in the relation:
\[ f(5 \cdot 3) = f(15) = 5 f(3) = 5 \cdot 2 = 10 \]

### Step 4: Continue applying the relation for higher powers of 5
We can continue applying the relation:
- \( f(5^2 \cdot 3) = f(75) = 5 f(15) = 5 \cdot 10 = 50 \)
- \( f(5^3 \cdot 3) = f(375) = 5 f(75) = 5 \cdot 50 = 250 \)
- \( f(5^4 \cdot 3) = f(1875) = 5 f(375) = 5 \cdot 250 = 1250 \)

### Conclusion
The maximum value of \( f(x) \) in the interval \( [5^4, 5^5] \) is \( 1250 \).

### Final Answer
The maximum value of \( f(x) \) in the interval \( [5^4, 5^5] \) is \( \boxed{1250} \).